# Fundamental theorems of welfare economics

There are two fundamental theorems of welfare economics. The First Theorem states that a market will tend toward a competitive equilibrium that is weakly Pareto optimal when the market maintains the following three attributes:[1]

1. Complete markets as no transaction costs and because of this each actor also has perfect information.

2. Price-taking behavior as no monopolists and easy entry and exit from a market.

Furthermore, the First Theorem states that the equilibrium will be fully Pareto optimal with the additional condition of:

3. Local nonsatiation of preferences as for any original bundle of goods there is another bundle of goods arbitrarily close to the original bundle, but that is preferred.

The Second Theorem states that out of all possible Pareto optimal outcomes one can achieve any particular one by enacting a lump-sum wealth redistribution and then letting the market take over.

## Implications of the First Theorem

The First Theorem is often taken to be an analytical confirmation of Adam Smith's "invisible hand" hypothesis, namely that competitive markets tend toward an efficient allocation of resources. The theorem supports a case for non-intervention in ideal conditions: let the markets do the work and the outcome will be Pareto efficient. However, Pareto efficiency is not necessarily the same thing as desirability; it merely indicates that no one can be made better off without someone being made worse off. There can be many possible Pareto efficient allocations of resources and not all of them may be equally desirable by society.[2]

This appears to make the case that intervention has a legitimate place in policy redistributions can allow us to select from all efficient outcomes for one that has other desired features, such as distributional equity. The shortcoming is that for the theorem to hold, the transfers have to be lump-sum and the government needs to have perfect information on individual consumers' tastes as well as the production possibilities of firms. An additional mathematical condition is that preferences and production technologies have to be convex.[3]

### Proof of the First Theorem

The first fundamental theorem was first demonstrated graphically by economist Abba Lerner and mathematically by economists Harold Hotelling, Oskar Lange, Maurice Allais, Lionel McKenzie, Kenneth Arrow and Gérard Debreu. The theorem holds under general conditions.[3]

The formal statement of the theorem is as follows: If preferences are locally nonsatiated, and if is a price equilibrium with transfers, then the allocation is Pareto optimal. An equilibrium in this sense either relates to an exchange economy only or presupposes that firms are allocatively and productively efficient, which can be shown to follow from perfectly competitive factor and production markets.[3]

Given a set of types of goods we work in the real vector space over , and use boldface for vector valued variables. For instance, if then would be a three dimensional vector space and the vector would represent the bundle of goods containing one unit of butter, 2 units of cookies and 3 units of milk.

Suppose that consumer i has wealth such that where is the aggregate endowment of goods (i.e. the sum of all consumer and producer endowments) and is the production of firm j.

Preference maximization (from the definition of price equilibrium with transfers) implies (using to denote the preference relation for consumer i):

if then

In other words, if a bundle of goods is strictly preferred to it must be unaffordable at price . Local nonsatiation additionally implies:

if then

To see why, imagine that but . Then by local nonsatiation we could find arbitrarily close to (and so still affordable) but which is strictly preferred to . But is the result of preference maximization, so this is a contradiction.

An allocation is a pair where and , i.e. is the 'matrix' (allowing potentially infinite rows/columns) whose ith column is the bundle of goods allocated to consumer i and is the 'matrix' whose jth column is the production of firm j. We restrict our attention to feasible allocations which are those allocations in which no consumer sells or producer consumes goods which they lack, i.e.,for every good and every consumer that consumers initial endowment plus their net demand must be positive similarly for producers.

Now consider an allocation that Pareto dominates . This means that for all i and for some i. By the above, we know for all i and for some i. Summing, we find:

.

Because is profit maximizing, we know , so . But goods must be conserved so . Hence, is not feasible. Since all Pareto-dominating allocations are not feasible, must itself be Pareto optimal.[3]

Note that while the fact that is profit maximizing is simply assumed in the statement of the theorem the result is only useful/interesting to the extent such a profit maximizing allocation of production is possible. Fortunately, for any restriction of the production allocation and price to a closed subset on which the marginal price is bounded away from 0, e.g., any reasonable choice of continuous functions to parameterize possible productions, such a maximum exists. This follows from the fact that the minimal marginal price and finite wealth limits the maximum feasible production (0 limits the minimum) and Tychonoff's theorem ensures the product of these compacts spaces is compact ensuring us a maximum of whatever continuous function we desire exists.

## Proof of the second fundamental theorem

The Second Theorem formally states that, under the assumptions that every production set is convex and every preference relation is convex and locally nonsatiated, any desired Pareto-efficient allocation can be supported as a price quasi-equilibrium with transfers.[3] Further assumptions are needed to prove this statement for price equilibria with transfers.

The proof proceeds in two steps: first, we prove that any Pareto-efficient allocation can be supported as a price quasi-equilibrium with transfers; then, we give conditions under which a price quasi-equilibrium is also a price equilibrium.

Let us define a price quasi-equilibrium with transfers as an allocation , a price vector p, and a vector of wealth levels w (achieved by lump-sum transfers) with (where is the aggregate endowment of goods and is the production of firm j) such that:

i. for all (firms maximize profit by producing )
ii. For all i, if then (if is strictly preferred to then it cannot cost less than )
iii. (budget constraint satisfied)

The only difference between this definition and the standard definition of a price equilibrium with transfers is in statement (ii). The inequality is weak here ( ) making it a price quasi-equilibrium. Later we will strengthen this to make a price equilibrium.[3] Define to be the set of all consumption bundles strictly preferred to by consumer i, and let V be the sum of all . is convex due to the convexity of the preference relation . V is convex because every is convex. Similarly , the union of all production sets plus the aggregate endowment, is convex because every is convex. We also know that the intersection of V and must be empty, because if it were not it would imply there existed a bundle that is strictly preferred to by everyone and is also affordable. This is ruled out by the Pareto-optimality of .

These two convex, non-intersecting sets allow us to apply the separating hyperplane theorem. This theorem states that there exists a price vector and a number r such that for every and for every . In other words, there exists a price vector that defines a hyperplane that perfectly separates the two convex sets.

Next we argue that if for all i then . This is due to local nonsatiation: there must be a bundle arbitrarily close to that is strictly preferred to and hence part of , so . Taking the limit as does not change the weak inequality, so as well. In other words, is in the closure of V.

Using this relation we see that for itself . We also know that , so as well. Combining these we find that . We can use this equation to show that fits the definition of a price quasi-equilibrium with transfers.

Because and we know that for any firm j:

for

which implies . Similarly we know:

for

which implies . These two statements, along with the feasibility of the allocation at the Pareto optimum, satisfy the three conditions for a price quasi-equilibrium with transfers supported by wealth levels for all i.

We now turn to conditions under which a price quasi-equilibrium is also a price equilibrium, in other words, conditions under which the statement "if then " imples "if then ". For this to be true we need now to assume that the consumption set is convex and the preference relation is continuous. Then, if there exists a consumption vector such that and , a price quasi-equilibrium is a price equilibrium.

To see why, assume to the contrary and , and exists. Then by the convexity of we have a bundle with . By the continuity of for close to 1 we have . This is a contradiction, because this bundle is preferred to and costs less than .

Hence, for price quasi-equilibria to be price equilibria it is sufficient that the consumption set be convex, the preference relation to be continuous, and for there always to exist a "cheaper" consumption bundle . One way to ensure the existence of such a bundle is to require wealth levels to be strictly positive for all consumers i.[3]

Because of welfare economics' close ties to social choice theory, Arrow's impossibility theorem is sometimes listed as a third fundamental theorem.[4]

The ideal conditions of the theorems, however are an abstraction. The Greenwald-Stiglitz theorem, for example, states that in the presence of either imperfect information, or incomplete markets, markets are not Pareto efficient. Thus, in real world economies, the degree of these variations from ideal conditions must factor into policy choices.[5] Further, even if these ideal conditions hold, the First Welfare Theorem fails in an overlapping generations model.

## References

1. http://web.stanford.edu/~hammond/effMktFail.pdf
2. Stiglitz, Joseph E. (1994), Whither Socialism?, MIT Press, ISBN 0-262-69182-5
3. Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995), "Chapter 16: Equilibrium and its Basic Welfare Properties", Microeconomic Theory, Oxford University Press, ISBN 0-19-510268-1
• Feldman, Allan M. (2008), "Welfare Economics", The New Palgrave: A Dictionary of Economics (online ed.), 4, pp. 889–95, retrieved 9 June 2014
4. Stiglitz, Joseph E. (March 1991), "The Invisible Hand and Modern Welfare Economics", NBER Working Paper No. W3641, doi:10.3386/w3641